orthocentrum

An orthocentrum, more commonly called the orthocenter, is the common intersection point of the three altitudes of a triangle. For triangle ABC, the altitude from A is the line through A perpendicular to BC, the altitude from B is through B perpendicular to CA, and the altitude from C is through C perpendicular to AB. These three lines are concurrent at a single point, designated H, the orthocenter.

The location of the orthocenter depends on the type of triangle. In an acute triangle, H lies inside the triangle. In a right triangle, H coincides with the vertex at the right angle. In an obtuse triangle, H lies outside the triangle.

Key properties include relations with the circumradius R: AH = 2R cos A, BH = 2R cos B, CH = 2R cos C, where A, B, C are the triangle’s angles. The orthocenter, the circumcenter O, and the centroid G are collinear on the Euler line, and the center of the nine-point circle is the midpoint of OH. The reflection of H across any side of the triangle lies on the circumcircle. In an equilateral triangle, H coincides with the centroid, circumcenter, and incenter.

The orthocenter is a standard object of study in Euclidean triangle geometry and appears in various classical results, constructions, and proofs.